Harmonic balance methods applied to computational fluid dynamics problems

In this paper, we briefly review the classical harmonic balance method, and describe the adaptation of the method required for its application to computational fluid dynamics models of unsteady time periodic flows. We describe several variations of the method including a classical balancing method with pseudo time relaxation, the nonlinear frequency domain form and the time spectral form. We show that the maximum stable CourantFriedrichsLewy (CFL) number for explicit schemes is dependent on the grid reduced frequency, a non-dimensional parameter that depends on the cell size, characteristic wave speed, and the highest frequency retained in the harmonic balance analysis. We apply the harmonic balance methods to several nonlinear unsteady flow problems and show that even strongly nonlinear flows can be modelled accurately with a small number of harmonics retained in the model.